Optimal. Leaf size=95 \[ \frac {2 b^3 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 b \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}} \]
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Rubi [A] time = 0.07, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 3769, 3771, 2639} \[ \frac {2 b^3 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 b \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2639
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x)}{\sqrt {b \sec (c+d x)}} \, dx &=b^4 \int \frac {1}{(b \sec (c+d x))^{9/2}} \, dx\\ &=\frac {2 b^3 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {1}{9} \left (7 b^2\right ) \int \frac {1}{(b \sec (c+d x))^{5/2}} \, dx\\ &=\frac {2 b^3 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 b \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac {7}{15} \int \frac {1}{\sqrt {b \sec (c+d x)}} \, dx\\ &=\frac {2 b^3 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 b \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac {7 \int \sqrt {\cos (c+d x)} \, dx}{15 \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}\\ &=\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 b^3 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 b \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 70, normalized size = 0.74 \[ \frac {4 (33 \sin (c+d x)+5 \sin (3 (c+d x))) \cos (c+d x)+\frac {336 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\sqrt {\cos (c+d x)}}}{360 d \sqrt {b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (d x + c\right )} \cos \left (d x + c\right )^{4}}{b \sec \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x + c\right )^{4}}{\sqrt {b \sec \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.99, size = 328, normalized size = 3.45 \[ \frac {2 \left (21 i \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}-21 i \cos \left (d x +c \right ) \sin \left (d x +c \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-5 \left (\cos ^{6}\left (d x +c \right )\right )+21 i \sin \left (d x +c \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-21 i \sin \left (d x +c \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-2 \left (\cos ^{4}\left (d x +c \right )\right )-14 \left (\cos ^{2}\left (d x +c \right )\right )+21 \cos \left (d x +c \right )\right ) \sqrt {\frac {b}{\cos \left (d x +c \right )}}}{45 d \sin \left (d x +c \right ) b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x + c\right )^{4}}{\sqrt {b \sec \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (c+d\,x\right )}^4}{\sqrt {\frac {b}{\cos \left (c+d\,x\right )}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^{4}{\left (c + d x \right )}}{\sqrt {b \sec {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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